A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Independence of two random variables derived from a Brownian motion

If $X = B_1 + B_3 - B_2$ and $Y = B_1 - B_3 + B_2$ Where $B_t$ is Brownian Motion for $t \geq 0$ And I want to state with certainty whether $X$ and $Y$ are indep or not, do I simply just ...
3
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1answer
30 views

Distribution of Brownian Motion help

If $X = \frac{B_1 - B_3 + B_2}{\sqrt{2}}$ Where $B_t$ is brownian motion at time $t$. And I want to find the the distribution of $X$, how would I do so? $E[X] = 0$ is fairly straight forward. For ...
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0answers
11 views

Girsanov's theorem and absolutely continuous restrictions

Let $W$ be a Brownian motion on some probability space $(\Omega, \mathcal{F}, P)$. Let $\mathbb{F}^W$ be the fi ltration generated by $W$ and let $X$ be a process that is progressively measurable ...
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16 views

conditional expectation of a sum of two independent Ornstein-Uhlenbeck type processes

Consider two independent processes of discrete Ornstein-Uhlenbeck type, $X_t$ and $Y_t$: \begin{eqnarray*} dX_t&=&\theta_x(\bar{X}-X_t) + \sigma_xdB_{xt}\\ dY_t&=&\theta_y(\bar{Y}-Y_t) ...
0
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1answer
14 views

Exponential Martingales - Properties

This question relates to the exponential martingale, \begin{align} Y(t) = \exp\left(-\int_{0}^{t} \lambda(s)\,dW(s) - \tfrac{1}{2} \int_{0}^{t} \lambda^2(s)\,ds \right) \end{align} and specifically ...
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13 views

Stationary distribution of a CTMC

We are talking about Continuous time Markov chains in class and we are supposed to show that for all s,t > 0: $\mu (t+s)= \mu (t)*P(s)$ and $\mu_i (t+s) = $$ \sum\limits_{j \in E}\mu_j(t)P_{ji}(s)$ ...
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10 views

Multivariate Laplace transform of independant Poisson processes

Let $W_1$ and $W_2$ be independant Poisson processes on $S$ with mean measures $\rho_1$ and $\rho_2$. Let $$w_1^\ast=\sum_{X\in W_1} X$$ and $$w_2^\ast = \sum_{X\in W_2} X$$ Reference: Kingman's ...
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23 views

Why do we need to declare a probability measure for the definition of stochastic processes?

Let $(\Omega,\mathcal{A})$ be a measurable space $E$ be a Polish space and $\mathcal{E}$ be the Borel-$\sigma$-algebra on $E$ $I\subseteq\mathbb{R}$ $X_t$ be measurable with respect to ...
2
votes
1answer
34 views

Martingale representation theorem application

Let $X = exp(W_{T/2}+W_T)$. I try to figure the adapted process $g(s)$ such that according to the MRT we have $$X = \mathbb{E}[X]+\int^T_0 g_s dW_s$$. I can figure out $X = exp(2W_{T/2}+W_{T-T/2})$ ...
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22 views

p.d.f. of a position variable from stochastic velocity p.d.f.

I have a stochastic process, $v(t)$, that represents a velocity, and has a known probability distribution function $f(x,t)$ which is time-varying. I am interested to acquire a probability ...
0
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0answers
27 views

Does this sequence converge? If yes, what is the limit?

Assume $\{k_n\}_{n\geq 0}$ a sequence of natural numbers such that $k_0=0$, $k_n\leq k_{n+1}\leq k_n+1$, and $\lim_{n\rightarrow\infty} \frac{k_n}{n}=\alpha\in(0,1)$. So $\{k_n\}$ is an ...
1
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1answer
24 views

proving independence of stochastic integrals

Does anyone know how to show that the stochastic integrals \begin{equation} \bigg\{ \int_0^1 \cos \Big[ (n- \frac{1}{2}) \pi t \Big] \,dW_t \bigg\}_{n \in \mathbb{N}} \end{equation} are ...
0
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1answer
15 views

Distribution of $\int^T_t \sigma (T-u)dW_u$ where $W_t$ is a Brownian motion

I am trying to find the distribution of $\int^T_t \sigma (T-u)dW_u$ where $W_t$ is a Brownian motion. One (very hand-wavey) way is to assume a priori that it is Normally distributed. Then one can ...
1
vote
1answer
13 views

Asymptotic stopping time for a ball-drawing problem

Take two different boxes, one with $N$ red balls and one with $N$ blue balls. Remove balls one at a time from either box with equal probability. When only one color is left, the (expected value of ...
1
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1answer
13 views

Construction of Probability Generating Function in Branching Process?

So I'm trying to construct a probability generating function for the following scenario: 1/5 of a rabbit population does not reproduce. 4/5 have 3 offspring each, and the probability of male or ...
1
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1answer
22 views

Properties of brownian motion

I was doing some revision and had an admittedly elementary question. My lecture notes say, the following are properties of Brownian Motion {$B_t$} (Normal or Gaussian increments) For all $s < t, ...
2
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1answer
28 views

Exponential Martingales

This is a two-part question concerning exponential martingales. It is stated that an application of Ito's lemma to \begin{align} \rho_t = \exp\left[-\int_{0}^{t} \lambda_s\,dW_s - ...
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1answer
29 views

Convergence of exponential Brownian martingale to zero almost surely

Define the exponential Brownian martingale as $N_t = \exp\left\{a W_t - \frac12 a^2 t \right\}$ which is a martingale with respect to the natural filtration of $W$ which stands for a standard Brownian ...
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27 views

Is there a name for this stochastic process?

Let $(\Omega,\mathscr{F},P)$ be a probability space and $\{X_n\}_{n\geq 1}$ be a stochastic process. Assume each $X_n$ only takes two values $0$ or $1$, i.e., $X_n:\Omega\rightarrow \{0,1\}$. Of ...
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0answers
12 views

Poisson Processes question

. Let {N(t) : t ≥ 0} be a Poisson process with rate λ > 0. Let Y be a random variable independent of (N(t)), such that Y = 1 with probability 1/2 and Y = −1 with probability 1/2. We define the new ...
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1answer
11 views

Poisson Processes - What is the distribution of the number of arrivals $Z$ happening in the random interval of time $[0,T]$?

Let $\{N(t) : t \geq 0\}$ be a Poisson process with rate $λ$, and $Z$ represent the number of arrivals in the interval of time $[0,t]$. Let $T$ be a random variable, exponentially distributed with ...
0
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1answer
30 views

Distribution of a transformed Brownian motion

Let $W$ be a standard Brownian motion. From an earlier proven result I know that $N_t = \exp\left\{a W_t - \frac12 a^2 t \right\}$ defines a martingale on the natural filtration of $W$ for all $a \in ...
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0answers
11 views

Book recommendation needed: asymptotic behavior of non-stationary Markov chain

Is there any stochastic process textbook which covers some standard results for non-stationary Markov chain? For my purpose, countable state space is enough. Thanks!
0
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1answer
28 views

Calculate $\mathbb{E}(T^2)$ and $\mathbb{E}(\int_0^T X_s \,d s)$ for exit time $T$ of Brownian motion $(X_t)_{t \geq 0}$

Let $T$ be the exit time of from the interval $[-b,a]$ of a standard Brownian Motion $X_t$, then how would we go about calculating the following two expectations: $E[T^2]$ (and) $E[\int_0^T X_tds]$? ...
2
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0answers
29 views

Measurability of the event that Brownian motion hits a given set

Let $W$ be a Brownian motion in $\mathbb{R}^{2}$ on a probability space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ . Let us assume $\mathcal{F}$ is the sigma-algebra on the path space ...
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0answers
7 views

Point processes that are not Cox?

Can some provide examples of point processes that are not Cox? A Cox process is a doubly stochastic poisson process with random intensity.
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1answer
65 views

Prove $\mathbb{P} \left( \sup_{t \geq 0} M_t > x \mid \mathcal{F}_0 \right) = 1 \wedge \frac{M_0}{x}$ for a martingale $(M_t)_{t \geq 0}$

Let $M$ be a positive, continuous martingale that converges a.s. to zero as $t$ tends to infinity. I now want to prove that for every $x>0$ $$ P\left( \sup_{t \geq 0 } M_t > x \mid \mathcal{F}_0 ...
1
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1answer
43 views

Supremal distribution of positive continuous martingale, which converges to zero a.s.

So the question is as follows: Let $M$ be a positive continous martingale, converging a.s. to zero as $t \rightarrow \infty$. Prove that for every $x>0$: $\mathbb{P}\{\sup_{\{t \geq 0 \}} M_t > ...
0
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0answers
12 views

Relationship between distributions of correlations $\rho(X^1,Y^1)$ and $\rho(X^2,Y^2)$ if $X^2=WX^1$, $Y^2=WY^1$ and $W$ is a known stochastic matrix?

I have been stacked for a while with the following problem: Consider two samples of iid observations $X^1=\{X_1^1,\dots,X_n^1\}$ and $Y_1=\{Y_1^1,\dots,Y_n^1\}$ where $X_i^1 \sim \mathcal{N}(0,1)$ and ...
3
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1answer
36 views

Reference request for stochastic process

I studied the book, "Probability, Random Variables and Random Signal Principles" by Peyton Peebles. And I am a little bit familiar with statistical analysis like signal estimation and detection. In ...
3
votes
1answer
33 views

If two stochastic processes are modifications of each other and almost surely continuous from the right, then they are undistinguishable

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $I\subseteq\mathbb{R}$ $E$ be a metric space and $\mathcal{E}:=\mathcal{B}(E)$ be the Borel-$\sigma$-algebra on $E$ $X:=(X_t)_{t\in ...
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0answers
16 views

Defining a stochastic process indexed by angle

I have a random closed curve of the form $(\theta,r_\theta)$, where $\theta\in [0,2\pi]$, is the counter clockwise angle from the x-axis and $r_\theta$ is the radial distance from the origin ...
0
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1answer
33 views

Adaptive gambler's ruin problem

Suppose in the gambler's ruin problem that the probability of winning a bet depends on the gambler's present fortune. Specifically, suppose that $p_{i}$ is the probability that the gambler wins a bet ...
-1
votes
1answer
28 views

Men and Women enter a supermarket according to independent poisson process (stochastic process) [on hold]

Men and Women enter a supermarket according to independent poisson processes having respective rates of two and four per minute. a) Starting at an arbitrary time, what is the probability that at ...
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0answers
12 views

Variance of interarrival time of events [on hold]

As shown in the figure, in this problem, there are three types of events where events of each type occur independently. The inter-arrival time distribution between events of the same type is an ...
2
votes
1answer
56 views

Generating the Borel $\sigma$-algebra on $C([0,1])$

We put $S=C([0,1])$ (the collection of continuous real functions on $[0,1]$), equipped with the metric $d(f,g)=\sup_{x\in[0,1]}|f(x)-g(x)|$, and let $\mathcal{B}(S)$ be the Borel $\sigma$-algebra on ...
2
votes
2answers
47 views

How to prove $E[e^{e^y}]=\infty$? y is a normal random variable

The question is, given $Y\sim N(\mu,\sigma^2)$, how to prove$E[e^{e^Y}]=\infty$? I tried to look Y as some kind of Ito's process and apply Ito's formula to it but it doesn't make sense. Next I tried ...
2
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1answer
37 views

Distribution of $\| W_t \|^2_{L^2([0,T])}$

Motivation: consider the SDE $$dX_t = b(X_t) dt + \sqrt{\varepsilon} dW_t. \tag1$$ Consider the action, defined by $$S(\phi)=\int_0^T |\phi'(t)-b(\phi(t))|^2 dt$$ if $\phi \in H^1([0,T])$ and ...
2
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1answer
51 views

Show that a Markov Chain is ergodic

Let $Y_n$ be iid random variables with values 1,2,3..n so that $P[Y_i=j]=p_j>0$, where $i\leq1$ and $1\leq j\leq n$. I think I managed to show that $Y_n$ is a Markov chain using the definition, ...
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1answer
35 views

Stronger version of Markov Chain

I have just started looking into the concept of Markov chains and I was wondering if anyone could help me with this problem. Let $X_1, X_2, ...$ be a Markov chain with the state space $S$. I need ...
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1answer
31 views

Process adapted to Filtration [closed]

Here is the definition I have been given : A process $(X_t)$ is adapted to a filtration $(\mathcal F_t)$ if $X_t$ is $F_t$ measurable, for all t > 0 , i.e : $X_t^-1 (\mathcal B)$ belong to ...
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0answers
22 views

Convergence in distribution of BM started in (x,y) to BM started in (0,0)

Let $B$ be a Brownian motion in $\mathbb{R}^{2}$ . Let $\mathbb{P}_{(x,y)}$ denote a probability measure under which $B$ is started at $(x,y)$ . Is it true in general that, for measurable set ...
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0answers
22 views

Simple Symmetric Random Walk [closed]

Use Hint: Show first that for any random variable N with range {0,1,...},
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1answer
35 views

Example of a white noise series that is not a martingale difference series with respect to its natural filtration

For a homework exercise, I am asked to find an example of a white noise series that is not a martingale difference series with respect to its natural filtration. Does anyone know an example? I read ...
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17 views

How to get the PDF for $p(a)$ $p(b)$ by using convolution?

Suppose there are three random variable $a$, $b$, $c$, and the PDF for each are $p(a)\ p(b)$ and $p(c) $ Also, $c$ = $a$ + $b$, $a$ and $b$ are two independent variable. and$$p(c)= \begin{cases} ...
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1answer
31 views

Applying the martingale representation theorem

I'm having trouble applying the martingale representation theorem to examples of Brownian martingales $M$ and contruct a process $X$ such that if we have a Brownian motion $W$ then $M= X \cdot W$. ...
2
votes
1answer
28 views

Stochastic kernel as linear operator

Let $K$ be a stochastic kernel for a set $S$ equipped with a countably generated $\sigma$-Algebra $B(S)$, i.e. $K:S\times B(S)\rightarrow [0,1]$ such that $K(\cdot,A)$ is a measurable function for ...
0
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1answer
19 views

Finding the expectation and variance of a stochastic process

Let $X_0, \ldots$ be i.i.d. $\mathbb{P}\{X_i = -1\} = \mathbb{P}\{X_i = 1\} = 1 / 2$. Given $a, b \in \mathbb{R}, |b| < 1$, consider the stochastic process $W_k$ defined as $$ W_0 = a X_0\\ W_k = b ...
0
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0answers
26 views

Two-state Markov Chains

If I have a two-state Markov chian $V(t)$ with transition probabilities: $P_{00}(t)=(1-\pi) + \pi e^{-\tau t}$ $P_{01}(t)= \pi - \pi e^{-\tau t}$ $P_{10}(t)=(1-\pi) - (1-\pi)e^{-\tau t}$ ...
3
votes
1answer
43 views

Does the power spectral density vanish when the frequency is zero for a zero-mean process?

A wide-sense stationary random time series $\zeta(t)$ is characterized by its mean value and its autocovariance function, which in the Wiener–Khinchin theorem is equivalent to the Fourier transform of ...